SL 2.8—Reciprocal and simple rational functions, equations of asymptotes

Reciprocal Function: The reciprocal function is defined as $f(x)=\frac{1}{x}$, where $x\ne 0$. Its domain and range are both $\mathbb{R}\setminus \{0\}$ (all real numbers except zero).

The graph of $f(x)=\frac{1}{x}$ is called a rectangular hyperbola and consists of two separate branches:

• A branch in the first quadrant (both $x$ and $y$ positive)
• A branch in the third quadrant (both $x$ and $y$ negative)

As $x$ grows very large (positive or negative), $y$ approaches zero but never reaches it. As $x$ approaches zero from either side, $|y|$ grows without bound.